On Prime Reciprocals in the Cantor Set
Abstract
The middle-third Cantor set C3 is a fractal consisting of all the points in [0, 1] which have non-terminating base-3 representations involving only the digits 0 and 2. It is easily shown that the reciprocals of all prime numbers p > 3 satisfying an equation of the form 2p + 1 = 3q belong to C3. Such prime numbers have base-3 representations consisting of a contiguous sequence of 1's and are known as base-3 repunit primes. It is natural to ask whether all prime numbers with reciprocals in C3 satisfy this equation. In this paper we show that the answer is no, but all primes with reciprocals in C3 do satisfy a closely related equation of the form 2pK + 1 = 3q. The base-3 repunit primes are thus shown to be a special case corresponding to K = 1.